In the Say Anything song "Death for my Birthday" the narrator spends the entirety of his (wonderful) life praying for death and being miserable. When he finally gets what he wants, he has a moment of reflection before he dies and wishes he had more time to enjoy all of the things he had.
Despite not being program music, Havergal Brian's "Gothic" Symphony has a sort of Bittersweet Ending.
Martina McBride's Concrete Angel definitely has a bittersweet ending (especially if you watch the music video). The poor abused girl ends up dying, but she flies up to a place where she's loved. The ending of the music video shows her little ghost self playing with a group of other children.
The Dream of Gerontius ends with the Angel dutifully delivering the Soul to Purgatory for cleansing. It's not as bad as Hell - there are other angels there whose sole duty, willingly undertaken, is to make the experience less unpleasant, and eventual release to eternal bliss is absolutely guaranteed - but Gerontius still has to be left behind as the Angel returns to his place in the courts of glory; and the song of the souls in Purgatory is, at best, resigned.
Ludo's The Broken Bride the protagonist is reunited with his wife for One Last Day, but ultimately decides to die with her in the car crash, rather than living without her
